Understanding the Kalman Filter with a Simple Radar Example

Summary

Kalman Filter example Let us begin with a simple example: a one-dimensional radar that measures range and velocity by transmitting a pulse toward an aircraft and receiving the reflected echo. The time delay between pulse transmission and echo reception provides information about the aircraft range \(r\), and the frequency shift of the reflected echo provides information about the aircraft velocity \(v\) (Doppler effect). In this example, the system state is described by both the aircraft range \(r\) and velocity \(v\). We define the system state by the vector \(\boldsymbol{x}\), which includes both quantities: \[ \boldsymbol{x}=\left[\begin{matrix}r\\v\\\end{matrix}\right] \] We denote vectors by lowercase bold letters and matrices by uppercase bold letters. Because the system state includes more than one variable, we use linear algebra tools, such as vectors and matrices, to describe the mathematics of the Kalman Filter. If you are not comfortable with linear algebra, please review the One-Dimensional Kalman Filter section in the online tutorial or in the book. It presents the Kalman Filter equations and their derivation using high-school-level mathematics, along with four fully solved examples. Iteration 0 Filter initialization In this example, we will use the first measurement to initialize the Kalman Filter (for more information on initialization techniques and their impact on the Kalman Filter performance, refer to Chapter 21 of the book). At time \(t_0\), the radar measures a range of \(10,000m\) and a velocity of \(200m/s\). The measurements are denoted by the letter \(\boldsymbol{z}\). We stack the measurements into the measurement vector \(\boldsymbol{z}\): \[ \boldsymbol{z}_0=\left[\begin{matrix}10{,}000\\200\\\end{matrix}\right] \] The subscript \(0\) indicates time \(t_0\). The measurement does not reflect the exact system state. Measurements are corrupted by random noise; therefore, each measurement is a random variable. Can we trust this measurement? H...